Using the Formal Definition


Prove the limit statements in Exercises 37–50.


limx → √3 1/x² = 1/3

Verified step by step guidance

1. Understand the formal definition of a limit: For a function f(x) and a limit L, we say that lim(x → a) f(x) = L if for every ε > 0, there exists a δ > 0 such that 0 < |x - a| < δ implies |f(x) - L| < ε.
2. Identify the function and the limit in the problem: Here, f(x) = 1/x² and L = 1/3, with a = √3.
3. Set up the inequality based on the definition: We need to show that for every ε > 0, there exists a δ > 0 such that 0 < |x - √3| < δ implies |1/x² - 1/3| < ε.
4. Simplify the inequality |1/x² - 1/3| < ε: This can be rewritten as |(3 - x²)/(3x²)| < ε. Focus on finding a δ that satisfies this condition.
5. Choose δ appropriately: Analyze the expression |3 - x²| < 3εx² and find a δ that works for the given ε. Consider the behavior of the function around x = √3 to ensure the inequality holds.

Key Concepts

Limit Definition

The formal definition of a limit states that for a function f(x) to have a limit L as x approaches a value c, for every ε > 0, there exists a δ > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This definition is crucial for rigorously proving limit statements.

Continuous Functions

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. Understanding continuity helps in evaluating limits, especially when dealing with rational functions like 1/x², as it ensures that the function behaves predictably near the limit point.

Rational Functions

Rational functions are ratios of polynomials, and their limits can often be evaluated by direct substitution unless the function is undefined at that point. In this case, analyzing the behavior of 1/x² as x approaches √3 is essential for determining the limit, which simplifies to 1/(√3)² = 1/3.